Abstract : Stochasticity, nonlinearity and complexity can be found and used in many different fields, including the natural sciences such as mechanics, physics, biology, neuroscience as well as technology and engineering fields such as aeronautics, astronautics, information theory and computer science. The symposium focuses on the stochastic modeling and data analysis in nonlinear dynamical system. The contributions cover various fields such as Brownian motion, Levy process, and fractional Brownian motion et al and applications, data analysis methods and techniques combining complex systems science and machine learning. This symposium provides a forum to discuss science, strengthen relationships, create new contacts and gain a direct experience of new progresses in stochastic modeling and data analysis.
[02085] Almost sure averaging for fast-slow stochastic differential equations via controlled rough path
Format : Talk at Waseda University
Author(s) :
Bin PEI (Northwestern Polytechnical University)
Yong Xu (Northwestern polytechnical university)
Abstract : We apply the averaging method to a coupled system consisting of two differential equations which has a slow component driven by fractional Brownian motion (FBM) with the Hurst parameter $\frac13 < H_1 \leq \frac12$ and a fast component driven by additive FBM with the Hurst parameter $\frac13 < H_2 \leq \frac12$. The main purpose is to show that the slow component of such a couple system can be described by a stochastic differential equation with averaged coefficients. Our main result deals with an averaging procedure which proves that the slow component converges almost surely to the solution of the corresponding averaged equation using the approach of time discretization and controlled rough path. To do this we generate a stationary solution by a exponentially attracting random fixed point of the random dynamical system generated by the fast component.
[01488] Discrete-time Approximation of Partially Observed Stochastic Optimal Control Problem
Format : Talk at Waseda University
Author(s) :
Yunzhang Li (Fudan University)
Abstract : In this talk, we study a class of stochastic optimal control problems under partial observation by discrete-time control problems. To establish a convergence result, we adapt the weak convergence technique together with the notion of relaxed control rule. With a well chosen discrete-time control system, we provide an implementable numerical algorithm to approximate the value. We illustrate our convergence result by numerical experiments on a partially observed control problem in a linear quadratic setting.
[03981] Large deviations for a slow-fast McKean-Vlasov model with jumps
Format : Talk at Waseda University
Author(s) :
Xiaoyu YANG (Northwestern Polytechnical University)
Yong Xu (Northwestern polytechnical university)
Abstract : We aim to investigate large deviations for a slow-fast McKean-Vlasov system with jumps. Based on the variational framework of the McKean-Vlasov system with jumps, it is turned into weak convergence for the controlled system. Different from the general case, the controlled system is related to the distribution of the original system, which causes difficulties. To solve it, the combination of asymptotic of original system and averaging principle is employed efficiently.
[02182] Recent advances in stochastic nonlinear dynamics: modeling, data analysis
Format : Online Talk on Zoom
Author(s) :
Zi-Fei Lin (Xi 'an University of Finance and Economics)
Yan-Ming liang (Xi 'an University of Finance and Economics)
Jia-Li Zhao (Xi 'an University of Finance and Economics)
Jiao-Rui Li (Xi 'an University of Finance and Economics)
Kapitaniak Tomasz (Lodz University of Technology)
Abstract : Predicting strongly noise-driven dynamic systems has always been a difficult problem due to their chaotic properties. In this study, we investigated the prediction of dynamic systems driven by strong noise intensities, which proves that deep learning can be applied in diverse fields. This is the first study that uses deep learning algorithms to predict dynamic systems driven by strong noise intensities. We examined the effect of hyperparameters in deep learning and introduced an improved algorithm for prediction. Several numerical examples are presented to illustrate the performance of the proposed algorithm, including the Lorenz system and the Rossler system driven by noise intensities of 0.1, 0.5, 1, and 1.25. All the results suggest that the proposed improved algorithm is feasible and effective for predicting strongly noise-driven dynamic systems. Furthermore, the influences of the number of Neurons, the Spectral Radius, and the Regularization Parameters are discussed in detail. These results indicate that the performances of the machine learning techniques can be improved by appropriately constructing the neural networks.
Xiaocong Liu (Northwestern Polytechnical University)
Yue Zhao (Northwestern Polytechnical University)
Abstract : Generalized cell mapping method based on deep learning is proposed which can predict responses of dynamical systems from experimental data with part of information about physical model. This method trains the neural network model from a small amount of experimental data and obtains the potential dynamic model. The global characteristics of system are analyzed by GCM method. By introducing deconvolution layer and image super-resolution, the probability density function of stochastic dynamic system response is estimated.
[02190] Complex dynamics of a conceptual airfoil structure with consideration of extreme flight conditions
Format : Talk at Waseda University
Author(s) :
Qi Liu (Tokyo Institute of Technology)
Abstract : An aircraft in practice serves under extreme flight conditions, that will have a substantial impact on its flight safety. Understanding dynamics of airfoil structure of an aircraft subjected to severe load conditions is thus extremely valuable and necessary. In this study, we will explore the complicated dynamical behaviors of a conceptual airfoil excited by an external harmonic force and an extreme random load. Importantly, such an extreme random load is portrayed by a non-Gaussian Lévy noise with a heavy-tailed feature. We theoretically deduce amplitude-frequency equations associated with the deterministic airfoil system. We observe excellent agreements between the analytical solutions and the numerical ones, as well as bistable behaviors. Besides, the effects of the extreme random load on the airfoil system are thoroughly investigated. Interestingly, within the bistable regime, the extreme random load can lead to stochastic transition and stochastic resonance. Due to its heavy-tailed nature, the Lévy noise would increase the possibility of a highly unexpected stochastic transition behavior between desirable low-amplitude and catastrophic high-amplitude oscillations compared with the Gaussian scenario. Such vibration patterns might damage or destroy the airfoil structure, which will put an aircraft in great danger. All the findings would be helpful in ensuring the flight safety and enhancing the strength and reliability of airfoil structure operating at extreme flight conditions.
[02205] Pattern Dynamics of Higher Order Reaction-Diffusion network
Format : Talk at Waseda University
Author(s) :
Jianwei Shen (North China University of Water Resources and Electric Power)
Abstract : In this paper, we will investigate the pattern dynamics of higher order reaction-diffusion network by group interactions and prove the interplay between different orders of interaction can affact the emergence of turing patterns. Our results try to the mechanism of many body interaction on complex network.
[02243] Three occurrence mechanisms of extreme events in stochastic dynamical systems
Format : Online Talk on Zoom
Author(s) :
Yongge Li (Northwestern Polytechnical University)
Dan Zhao (Northwestern Polytechnical University)
Yong Xu (Northwestern polytechnical university)
Abstract : In this work, three mechanisms for the occurrence of extreme events in stochastic dynamical systems are given. Firstly, for systems with a bifurcation structure, if the difference of the branches at the bifurcation point is large, then a time-varying amplitude of the external periodic excitation is able to induce an extreme event. This is verified in the rolling motion of a ship system. Secondly, for systems with rare attractors, a random pulse excitation, such as Poisson white noise, is able to drive the system to escape from the basin of general attractor to that of rare attractor. However, the basin of rare attractor is so small that the system will go back to the general state immediately. Such a kind of transition is also extreme event. Finally, an extreme excitation can also generate extreme event, such as the Lévy noise. In such cases, it does not require much about the systems, but the extreme excitations work. These results provide theoretical guidance for further prediction and avoidance of extreme events.
[02092] Homogenization of the two dimensional singular polymer measure
Format : Online Talk on Zoom
Author(s) :
Huanyu Yang (Free University of Berlin)
Abstract : We consider the SDE on $[0,n]$:
\begin{equation}
\left\{
\begin{aligned}
dZ^n_t&=\nabla h(n-t,Z^n_t)dt+dB_t,\\
Z_0^n&=x,
\end{aligned}
\right.
\end{equation}
where $B$ is the standard Brownian motion and $h$ is the solution to KPZ equation driven by space-white noise $\xi$ on $\mathbb{T}^2$.
The law of $Z^n$ is the same as the law of the coordinate process under the sigular polymer measure $Q^n$ constructed by Cannizzaro and Chouk:
\begin{equation}
Q^n(d\omega)=\frac{1}{U(n)}\exp\left(\int_0^n\left(\xi\left(\omega\left(s\right)\right)-\infty\right)\right)\mathbb{W}(d\omega),
\end{equation}
where $\mathbb{W}$ is the Wiener measure on $C([0,\infty),\mathbb{R})$. By decomposing the drift term $h$ into a singular time-homogenious term and a smooth time-dependent term, we prove that $\left(\frac{1}{\sqrt{n}}Z^n_{nt}\right)_{t\in[0,1]}$ converges to a Brownian motion.
This is the joint work with Nicolas Perkowski.
